The present invention relates to antennas and resonators, and more specifically to the design of non-Euclidian antennas and non-Euclidian resonators.
Antenna are used to radiate and/or receive typically electromagnetic signals, preferably with antenna gain, directivity, and efficiency. Practical antenna design traditionally involves trade-offs between various parameters, including antenna gain, size, efficiency, and bandwidth.
Antenna design has historically been dominated by Euclidean geometry. In such designs, the closed antenna area is directly proportional to the antenna perimeter. For example, if one doubles the length of an Euclidean square (or xe2x80x9cquadxe2x80x9d) antenna, the enclosed area of the antenna quadruples. Classical antenna design has dealt with planes, circles, triangles, squares, ellipses, rectangles, hemispheres, paraboloids, and the like, (as well as lines). Similarly, resonators, typically capacitors (xe2x80x9cCxe2x80x9d) coupled in series and/or parallel with inductors (xe2x80x9cLxe2x80x9d), traditionally are implemented with Euclidian inductors.
With respect to antennas, prior art design philosophy has been to pick a Euclidean geometric construction, e.g., a quad, and to explore its radiation characteristics, especially with emphasis on frequency resonance and power patterns. The unfortunate result is that antenna design has far too long concentrated on the ease of antenna construction, rather than on the underlying electromagnetics.
Many prior art antennas are based upon closed-loop or island shapes. Experience has long demonstrated that small sized antennas, including loops, do not work well, one reason being that radiation resistance (xe2x80x9cRxe2x80x9d) decreases sharply when the antenna size is shortened. A small sized loop, or even a short dipole, will exhibit a radiation pattern of 1/2xcex and 1/4xcex, respectively, if the radiation resistance R is not swamped by substantially larger ohmic (xe2x80x9cOxe2x80x9d) losses. Ohmic losses can be minimized using impedance matching networks, which can be expensive and difficult to use. But although even impedance matched small loop antennas can exhibit 50% to 85% efficiencies, their bandwidth is inherently narrow, with very high Q, e.g., Q greater than 50. As used herein, Q is defined as (transmitted or received frequency)/(3 dB bandwidth).
As noted, it is well known experimentally that radiation resistance R drops rapidly with small area Euclidean antennas. However, the theoretical basis is not generally known, and any present understanding (or misunderstanding) appears to stem from research by J. Kraus, noted in Antennas (Ed. 1), McGraw Hill, New York (1950), in which a circular loop antenna with uniform current was examined. Kraus"" loop exhibited a gain with a surprising limit of 1.8 dB over an isotropic radiator as loop area fells below that of a loop having a 1 xcex-squared aperture. For small loops of area A less than xcex2/100, radiation resistance R was given by:   R  =      K    ·                  (                  A                      λ            2                          )            2      
where K is a constant, A is the enclosed area of the loop, and xcex is wavelength. Unfortunately, radiation resistance R can all too readily be less than 1 xcexa9 for a small loop antenna.
From his circular loop research Kraus generalized that calculations could be defined by antenna area rather than antenna perimeter, and that his analysis should be correct for small loops of any geometric shape. Kraus"" early research and conclusions that small-sized antennas will exhibit a relatively large ohmic resistance O and a relatively small radiation resistance R, such that resultant low efficiency defeats the use of the small antenna have been widely accepted. In fact, some researchers have actually proposed reducing ohmic resistance O to 0 xcexa9 by constructing small antennas from superconducting material, to promote efficiency.
As noted, prior art antenna and resonator design has traditionally concentrated on geometry that is Euclidean. However, one non-Euclidian geometry is fractal geometry. Fractal geometry may be grouped into random fractals, which are also termed chaotic or Brownian fractals and include a random noise components, such as depicted in FIG. 3, or deterministic fractals such as shown in FIG. 1C.
In deterministic fractal geometry, a self-similar structure results from the repetition of a design or motif (or xe2x80x9cgeneratorxe2x80x9d), on a series of different size scales. One well known treatise in this field is Fractals, Endlessly Repeated Geometrical Figures, by Hans Lauwerier, Princeton University Press (1991), which treatise applicant refers to and incorporates herein by reference.
FIGS. 1A-2D depict the development of some elementary forms of fractals. In FIG. 1A, a base element 10 is shown as a straight line, although a curve could instead be used. In FIG. 1B, a so-called Koch fractal motif or generator 20-1, here a triangle, is inserted into base element 10, to form a first order iteration (xe2x80x9cNxe2x80x9d) design, e.g., N=1. In FIG. 1C, a second order N=2 iteration design results from replicating the triangle motif 20-1 into each segment of FIG. 1B, but where the 20-1xe2x80x2 version has been differently scaled, here reduced in size. As noted in the Lauwerier treatise, in its replication, the motif may be rotated, translated, scaled in dimension, or a combination of any of these characteristics. Thus, as used herein, second order of iteration or N=2 means the fundamental motif has been replicated, after rotation, translation, scaling (or a combination of each) into the first order iteration pattern. A higher order, e.g., N=3, iteration means a third fractal pattern has been generated by including yet another rotation, translation, and/or scaling of the first order motif.
In FIG. 1D, a portion of FIG. 1C has been subjected to a further iteration (N=3) in which scaled-down versions 20-1xe2x80x3 of the triangle motif 20-1 have been inserted into each segment of the left half of FIG. 1C. FIGS. 2A-2C follow what has been described with respect to FIGS. 1A-1C, except that a rectangular motif 20-2 has been adopted, which motif is denoted 20-2xe2x80x2 in FIG. 2C, and 20-2xe2x80x3 in FIG. 2D. FIG. 2D shows a pattern in which a portion of the left-hand side is an N=3 iteration of the 20-2 rectangle motif, and in which the center portion of the figure now includes another motif, here a 20-1 type triangle motif, and in which the right-hand side of the figure remains an N=2 iteration.
Traditionally, non-Euclidean designs including random fractals have been understood to exhibit antiresonance characteristics with mechanical vibrations. It is known in the art to attempt to use non-Euclidean random designs at lower frequency regimes to absorb, or at least not reflect sound due to the antiresonance characteristics. For example, M. Schroeder in Fractals, Chaos, Power Laws (1992), W. H. Freeman, New York discloses the use of presumably random or chaotic fractals in designing sound blocking diffusers for recording studios and auditoriums.
Experimentation with non-Euclidean structures has also been undertaken with respect to electromagnetic waves, including radio antennas. In one experiment, Y. Kim and D. Jaggard in The Fractal Random Array, Proc. IEEE 74, 1278-1280 (1986) spread-out antenna elements in a sparse microwave array, to minimize sidelobe energy without having to use an excessive number of elements. But Kim and Jaggard did not apply a fractal condition to the antenna elements, and test results were not necessarily better than any other techniques, including a totally random spreading of antenna elements. More significantly, the resultant array was not smaller than a conventional Euclidean design.
Prior art spiral antennas, cone antennas, and V-shaped antennas may be considered as a continuous, deterministic first order fractal, whose motif continuously expands as distance increases from a central point. A log-periodic antenna may be considered a type of continuous fractal in that it is fabricated from a radially expanding structure. However, log periodic antennas do not utilize the antenna perimeter for radiation, but instead rely upon an arc-like opening angle in the antenna geometry. Such opening angle is an angle that defines the size-scale of the log-periodic structure, which structure is proportional to the distance from the antenna center multiplied by the opening angle. Further, known log-periodic antennas are not necessarily smaller than conventional driven element-parasitic element antenna designs of similar gain.
Unintentionally, first order fractals have been used to distort the shape of dipole and vertical antennas to increase gain, the shapes being defined as a Brownian-type of chaotic fractalg. See F. Landstorrer and R. Sacher, Optimisation of Wire Antennas, J. Wiley, New York (1985). FIG. 3 depicts three bent-vertical antennas developed by Landstorfer and Sacher through trial and error, the plots showing the actual vertical antennas as a function of x-axis and y-axis coordinates that are a function of wavelength. The xe2x80x9cEFxe2x80x9d and xe2x80x9cBFxe2x80x9d nomenclature in FIG. 3 refer respectively to end-fire and back-fire radiation patterns of the resultant bent-vertical antennas.
First order fractalg have also been used to reduce horn-type antenna geometry, in which a double-ridge horn configuration is used to decrease resonant frequency. See J. Kraug in Antennas, McGraw Hill, New York p1885). The use of rectangular, box-like, and triangular shapes as impedance-matching loading elements to shorten antenna element dimensions is also known in the art.
Whether intentional or not, such prior art attempts to use a quasi-fractal or fractal motif in an antenna employ at best a first order iteration fractal. By first iteration it is meant that one Euclidian structure is loaded with another Euclidean structure in a repetitive fashion, using the same size for repetition. FIG. 1C, for example, is not first order because the 20-1xe2x80x2 triangles have been shrunk with respect to the size of the first motif 20-1.
Prior art antenna design does not attempt to exploit multiple scale self-similarity of real fractals. This is hardly surprising in view of the accepted conventional wisdom that because such antennas would be anti-resonators, and/or if suitably shrunken would exhibit so small a radiation resistance R, that the substantially higher ohmic losses O would result in too low an antenna efficiency for any practical use. Further, it is probably not possible to mathematically predict such an antenna design, and high order iteration fractal antennas would be increasingly difficult to fabricate and erect, in practice.
FIGS. 4A and 4B depict respective prior art series and parallel type resonator configurations, comprising capacitors C and Euclidean inductors L. In the series configuration of FIG. 4A, a notch-filter characteristic is presented in that the impedance from port A to port B is high except at frequencies approaching resonance, determined by 1/ (LC).
In the distributed parallel configuration of FIG. 4B, a low-pass filter characteristic is created in that at frequencies below resonance, there is a relatively low impedance path from port A to port B, but at frequencies greater than resonant frequency, signals at port A are shunted to ground (e.g., common terminals of capacitors C), and a high impedance path is presented between port A and port B. Of course, a single parallel LC configuration may also be created by removing (e.g., short-circuiting) the rightmost inductor L and right two capacitors C, in which case port B would be located at the bottom end of the leftmost capacitor C.
In FIGS. 4A and 4B, inductors L are Euclidean in that increasing the effective area captured by the inductors increases with increasing geometry of the inductors, e.g., more or larger inductive windings or, if not cylindrical, traces comprising inductance. In such prior art configurations as FIGS. 4A and 4B, the presence of Euclidean inductors L ensures a predictable relationship between L, C and frequencies of resonance.
Thus, with respect to antennas, there is a need for a design methodology that can produce smaller-scale antennas that exhibit at least as much gain, directivity, and efficiency as larger Euclidean counterparts. Preferably, such design approach should exploit the multiple scale self-similarity of real fractals, including Nxe2x89xa72 iteration order fractals. Further, as respects resonators, there is a need for a non-Euclidean resonator whose presence in a resonating configuration can create frequencies of resonance beyond those normally presented in series and/or parallel LC configurations.
The present invention provides such antennas, as well as a method for their design.
The present invention provides an antenna having at least one element whose shape, at least is part, is substantially a deterministic fractal of iteration order Nxe2x89xa72. Using fractal geometry, the antenna element has a self-similar structure resulting from the repetition of a design or motif (or xe2x80x9cgeneratorxe2x80x9d) that is replicated using rotation, and/or translation, and/or scaling. The fractal element will have x-axis, y-axis coordinates for a next iteration N+1 defined by xN+1=f(xN, ybN) and yN+1=g(xN, yN, where xN, yN define coordinates for a preceding iteration, and where f(x,y) and g(x,y) are functions defining the fractal motif and behavior.
In contrast to Euclidean geometric antenna design, deterministic fractal antenna elements according to the present invention have a perimeter that is not directly proportional to area. For a given perimeter dimension, the enclosed area of a multi-iteration fractal will always be as small or smaller than the area of a corresponding conventional Euclidean antenna.
A fractal antenna has a fractal ratio limit dimension D given by log(L)/log(r), where L and r are one-dimensional antenna element lengths before and after fractalization, respectively.
According to the present invention, a fractal antenna perimeter compression parameter (PC) is defined as:       PC    =                  full        ⁢                  -                ⁢        sized        ⁢                  xe2x80x83                ⁢        antenna        ⁢                  xe2x80x83                ⁢        element        ⁢                  xe2x80x83                ⁢        length                    fractal        ⁢                  -                ⁢        reduced        ⁢                  xe2x80x83                ⁢        antenna        ⁢                  xe2x80x83                ⁢        element        ⁢                  xe2x80x83                ⁢        length                        where      :              
            ⁢      PC        =          A      ·              log        ⁢                  xe2x80x83                [                  N          ⁡                      (                          D              +              C                        )                          ]            
in which A and C are constant coefficients for a given fractal motif, N is an iteration number, and D is the fractal dimension, defined above.
Radiation resistance (R) of a fractal antenna decreases as a small power of the perimeter compression (PC), with a fractal loop or island always exhibiting a substantially higher radiation resistance than a small Euclidean loop antenna of equal size. In the present invention, deterministic fractals are used wherein A and C have large values, and thus provide the greatest and most rapid element-size shrinkage. A fractal antenna according to the present invention will exhibit an increased effective wavelength.
The number of resonant nodes of a fractal loop-shaped antenna according to the present invention increases as the iteration number N and is at least as large as the number of resonant nodes of an Euclidean island with the same area. Further, resonant frequencies of a fractal antenna include frequencies that are not harmonically related.
A fractal antenna according to the present invention is smaller than its Euclidean counterpart but provides at least as much gain and frequencies of resonance and provides essentially a 50xcexa9 termination impedance at its lowest resonant frequency. Further, the fractal antenna exhibits non-harmonically frequencies of resonance, a low Q and resultant good bandwidth, acceptable standing wave ratio (xe2x80x9cSWRxe2x80x9d), a radiation impedance that is frequency dependent, and high efficiencies. Fractal inductors of first or higher iteration order may also be provided in LC resonators, to provide additional resonant frequencies including non-harmonically related frequencies.
Other features and advantages of the invention will appear from the following description in which the preferred embodiments have been set forth in detail, in conjunction with the accompanying drawings.